The Weakest Link: A Nodal Tension Model for Local Network Resilience

Abstract: The resilience of networked systems, defined by their ability to withstand targeted disruptions between a source and a target, is a critical concern in fields from ecology to infrastructure management. While spectral methods offer global insights, characterising the specific vulnerability of targeted pathways requires a more direct approach. In this paper, we frame this problem of local resilience as a continuous $L_1$ Nodal Tension linear program, built upon the classic dual of the maximum $s-t$ flow problem. While operations research establishes that this LP recovers the combinatorial minimum cut, we explicitly leverage its continuous polyhedral structure to transition into differential graph geometry and representation learning. Our framework formalises the structural redundancy gap between $L_1$ cuts and $L_2$ conductance, proves that the continuous bottleneck isolates negative discrete Ricci curvature, and derives a Local Cheeger Inequality to bound Message Passing Neural Network (MPNN) over-squashing. Furthermore, we extract the Clarke subdifferential of the capacity, establishing the Nodal Tension LP as a structurally sparse and Lipschitz-robust differentiable layer. We validate these theoretical properties computationally against standard algorithms. We then apply our model to a real-world conservation problem: assessing the connectivity of a grizzly bear corridor in the Canadian Rocky Mountains. The analysis reveals a structurally counter-intuitive feature of the landscape: the corridor's weakest link is not a remote bottleneck, but the local perimeter of the source protected area itself. By formalising this "null signal" for a classic choke point through our commute time bounds and subgradient analysis, we demonstrate our model's utility in translating LP bounds and graph theory into physical diagnostics. Our work provides a continuous and differentiable characterisation of local network resilience, bridging classical graph cuts with gradient-based representation learning. The source code to reproduce all results in the paper is available at https://anonymous.4open.science/r/tmlr-ldnr.

Submission Type: Long submission (more than 12 pages of main content)

Assigned Action Editor: ~Baoxiang_Wang1

Submission Number: 7171